Integration by substitutionEdexcel International A Level Maths: Subtopic test

1. 1
   Let $I=\int 2x(x^2+3)^4\,dx$. Use the substitution $u=x^2+3$.

   (a)

   Express $I$ in terms of $u$.

   [1 mark]

   • A$\int 2u^4\,du$
   • B$\int u^4\,du$
   • C$\int 2x\,u^4\,du$
   • D$\int 4u^3\,du$

   (b)

   Hence find $I$.

   [1 mark]

   • A$\frac15u^5+c$
   • B$(x^2+3)^5+c$
   • C$\frac15(x^2+3)^5+c$
   • D$\frac25x(x^2+3)^5+c$

   (c)

   Hence find the exact value of $\int_0^1 2x(x^2+3)^4\,dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $I=\int x\sqrt{x-2}\,dx$. Use the substitution $u=x-2$.

   (a)

   Express $I$ in terms of $u$.

   [1 mark]

   • A$\int u\sqrt u\,du$
   • B$\int (u-2)\sqrt u\,du$
   • C$\int (u+2)\,du$
   • D$\int (u+2)\sqrt u\,du$

   (b)

   Find $\int (u+2)\sqrt u\,du$.

   [1 mark]

   • A$\frac25u^{5/2}+\frac43u^{3/2}+c$
   • B$\frac25u^{5/2}+\frac23u^{3/2}+c$
   • C$\frac52u^{5/2}+3u^{3/2}+c$
   • D$u^{3/2}+2u^{1/2}+c$

   (c)

   Hence find the exact value of $\int_2^6x\sqrt{x-2}\,dx$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $I=\int_0^4\frac{x}{\sqrt{2x+1}}\,dx$.

   (a)

   Using the substitution $u=2x+1$, show that $I=\frac14\int_1^9\left(u^{1/2}-u^{-1/2}\right)du$.

   [3 marks]

   (b)

   Hence find the exact value of $I$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y=\frac{3x}{\sqrt{x+1}}$ for $x\ge0$.

   (a)

   Use the substitution $u=\sqrt{x+1}$ to find the exact area of the region bounded by $C$, the $x$-axis and the line $x=3$.

   [6 marks]

   (b)

   The line $L$ has equation $y=\frac32x$. Show that $L$ meets $C$ at the origin and at the point $P$ where $x=3$, and find the exact area of the region between $C$ and $L$.

   [6 marks]

   Total for question 4: 12 marks

End of questions